QUESTION IMAGE
Question
the image shows a complex fraction: the numerator is (x + 4)/3 + 1/x, and the denominator is 3/x² - 1/3.
To simplify the complex fraction \(\frac{\frac{x + 4}{3}+\frac{1}{x}}{\frac{3}{x^{2}}-\frac{1}{3}}\), we follow these steps:
Step 1: Simplify the numerator
The numerator is \(\frac{x + 4}{3}+\frac{1}{x}\). To add these two fractions, we find a common denominator, which is \(3x\).
We can factor the numerator \(x^{2}+4x + 3\) as \((x + 1)(x + 3)\), so the numerator simplifies to \(\frac{(x + 1)(x + 3)}{3x}\).
Step 2: Simplify the denominator
The denominator is \(\frac{3}{x^{2}}-\frac{1}{3}\). To subtract these two fractions, we find a common denominator, which is \(3x^{2}\).
We can factor the numerator \(9 - x^{2}\) as a difference of squares: \(9 - x^{2}=(3 - x)(3 + x)\), so the denominator simplifies to \(\frac{(3 - x)(3 + x)}{3x^{2}}\).
Step 3: Divide the simplified numerator by the simplified denominator
Dividing the simplified numerator \(\frac{(x + 1)(x + 3)}{3x}\) by the simplified denominator \(\frac{(3 - x)(3 + x)}{3x^{2}}\) is equivalent to multiplying the numerator by the reciprocal of the denominator:
Step 4: Cancel common factors
We can cancel the common factors of \(3\), \(x\), and \((x + 3)\) (note that \((x + 3)\) and \((3 + x)\) are the same):
Final Answer
\(\boxed{\frac{x(x + 1)}{3 - x}}\)
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To simplify the complex fraction \(\frac{\frac{x + 4}{3}+\frac{1}{x}}{\frac{3}{x^{2}}-\frac{1}{3}}\), we follow these steps:
Step 1: Simplify the numerator
The numerator is \(\frac{x + 4}{3}+\frac{1}{x}\). To add these two fractions, we find a common denominator, which is \(3x\).
We can factor the numerator \(x^{2}+4x + 3\) as \((x + 1)(x + 3)\), so the numerator simplifies to \(\frac{(x + 1)(x + 3)}{3x}\).
Step 2: Simplify the denominator
The denominator is \(\frac{3}{x^{2}}-\frac{1}{3}\). To subtract these two fractions, we find a common denominator, which is \(3x^{2}\).
We can factor the numerator \(9 - x^{2}\) as a difference of squares: \(9 - x^{2}=(3 - x)(3 + x)\), so the denominator simplifies to \(\frac{(3 - x)(3 + x)}{3x^{2}}\).
Step 3: Divide the simplified numerator by the simplified denominator
Dividing the simplified numerator \(\frac{(x + 1)(x + 3)}{3x}\) by the simplified denominator \(\frac{(3 - x)(3 + x)}{3x^{2}}\) is equivalent to multiplying the numerator by the reciprocal of the denominator:
Step 4: Cancel common factors
We can cancel the common factors of \(3\), \(x\), and \((x + 3)\) (note that \((x + 3)\) and \((3 + x)\) are the same):
Final Answer
\(\boxed{\frac{x(x + 1)}{3 - x}}\)